Mastering Absolute Value GRE Questions: Tips and Strategies for Success

Absolute value is one of the most misunderstood topics in GRE Quantitative Reasoning. It’s a concept that may seem simple at first glance, but can lead to confusion when it’s embedded in equations or inequalities. Whether you’re struggling to get your head around it or just looking for a refresher, this guide will walk you through absolute value step by step. We’ll explore what absolute value is, how to express it, and how it plays a significant role in solving GRE-style problems.

What Is Absolute Value?

Absolute value is a fundamental mathematical concept, typically introduced at an early stage in algebra. In simple terms, the absolute value of a number is its distance from zero on the number line, regardless of the direction. This means that the absolute value of a number is always non-negative, whether the number itself is positive or negative.

For example, consider the number 50. Its absolute value is simply 50 because it is 50 units away from zero on the number line. Similarly, the absolute value of -50 is also 50 because -50 is also 50 units away from zero, but in the opposite direction. In mathematical terms, this can be written as:

• The absolute value of 50 is written as |50| = 50. 
• The absolute value of -50 is written as |-50| = 50. 

The key takeaway here is that the absolute value ignores the sign of a number and instead focuses on how far that number is from zero.

Why Does Absolute Value Matter for the GRE?

On the GRE, absolute value often appears in various forms, including equations and inequalities, and can be a source of confusion for many test-takers. While the concept itself is straightforward, understanding how to manipulate it in more complex scenarios is critical for answering quantitative questions correctly. In this article, we will break down how to handle absolute value equations, inequalities, and absolute value expressions, all of which are essential to achieving a high score on the quantitative section.

One of the first things to keep in mind is that no matter what absolute value is used for, the result will always be non-negative, unless the number is zero. The only time you encounter zero in absolute value is when the number inside the bars is exactly zero. For example:

• The absolute value of 0 is |0| = 0. 

In any case, understanding this core concept and its implications on algebraic expressions will help you navigate absolute value problems on the GRE more effectively.

Absolute Value Notation: How to Express It

Mathematically, absolute value is expressed using vertical bars around the number or expression in question. For instance, if you have the number 7, its absolute value is written as |7|, which equals 7. The same rule applies to negative numbers. For instance, |-7| equals 7.

This notation is critical because, as we move into more complex problems, recognizing how to handle absolute value notation correctly will be vital. For example, when solving equations or inequalities that involve absolute value, we will need to express the absolute value symbolically and use the right steps to eliminate or manipulate the bars.

If you are dealing with a variable instead of a constant number, say |x|, the absolute value notation tells you that x can be either positive or negative. This dual possibility is central when solving absolute value equations or inequalities.

Here are a few more examples of absolute value notation:

• |x| represents the absolute value of x, meaning x is either positive or negative. 
• If |a| = b, then a = b or a = -b, provided that b is non-negative. 

At this point, you should understand that absolute value notation is not just about representing a number; it also serves as a signal for handling different cases when solving equations or inequalities involving absolute values.

The Core Properties of Absolute Value

Understanding the core properties of absolute value is crucial for tackling GRE questions effectively. Below are the most important properties that you should be familiar with.

1. The Absolute Value of Zero

The absolute value of zero is zero itself:

|0| = 0.

This is an important property to remember because it serves as the baseline in absolute value problems. When an equation involves zero inside the absolute value bars, you are dealing with an equation that has no “direction” (positive or negative), making it simpler to solve.

2. The Absolute Value of Positive Numbers

The absolute value of any positive number is simply the number itself. That is:

|x| = x, where x > 0.

For example:

|5| = 5.

3. The Absolute Value of Negative Numbers

The absolute value of any negative number is the positive counterpart of that number. In other words:

|x| = -x, where x < 0.

For example:

|-5| = 5.

4. The Absolute Value of a Product

The absolute value of a product is the product of the absolute values of the individual factors. That is:

|ab| = |a| * |b|.

This property will come in handy when dealing with absolute value expressions in equations.

5. The Absolute Value of a Sum

The absolute value of a sum is generally not equal to the sum of the absolute values. That is:

|a + b| ≠ |a| + |b|.

For example:

|3 + (-5)| = | -2 | = 2, but |3| + |-5| = 3 + 5 = 8.

This is a critical point because it means that you cannot simply break down absolute value problems involving sums without applying the correct process.

6. The Triangle Inequality

One of the most important properties of absolute value is the triangle inequality. It states that:

|a + b| ≤ |a| + |b|.

This inequality is useful when working with more complex absolute value problems where you need to find bounds or limits of a quantity.

Applying Absolute Value to GRE Quantitative Problems

As you begin working on GRE quantitative problems that involve absolute values, remember that absolute value questions often fall into one of the following categories:

• Basic absolute value equations (e.g., |x| = 5) 
• Equations involving more complicated algebraic expressions (e.g., |2x – 3| = 7) 
• Absolute value inequalities (e.g., |x – 4| < 5) 
• Problems that require comparing quantities (e.g., comparing absolute values of expressions) 

The first step in solving these types of problems is to understand how to express absolute value properly. Then, you need to apply the appropriate strategies for solving the equations or inequalities step by step. As you work through more problems, you’ll become more comfortable using absolute value notation and strategies to simplify or solve these types of problems.

while absolute value is a seemingly simple concept, its application in GRE Quantitative Reasoning can become tricky. The key is understanding the definition, mastering the notation, and applying the core properties effectively to solve equations, inequalities, and comparison problems. In the next section, we will dive deeper into solving absolute value equations and offer tips for tackling these types of questions efficiently.

Solving Absolute Value Equations: Step-by-Step Guide

Now that we’ve discussed the basic concept of absolute value and its properties, it’s time to dive into how absolute value is applied to equations. This part of the guide will walk you through the process of solving absolute value equations, which are a key area of focus for the GRE Quantitative section. While solving absolute value equations may initially seem intimidating, understanding the process and using a systematic approach will make these problems much easier to handle.

Solving Simple Absolute Value Equations

The simplest form of an absolute value equation is one in which the absolute value of a single expression is set equal to a constant. For example:

|x| = 5

This equation means that the distance between x and 0 on the number line is 5 units, regardless of whether x is positive or negative. To solve this equation, you need to account for both possibilities:

• Case 1: x = 5 
• Case 2: x = -5 

This is because both 5 and -5 are exactly 5 units away from zero on the number line. The solution to this equation is:

x = 5 or x = -5

So, the general rule when solving absolute value equations of this form (|x| = n) is:

• x = n or x = -n (provided n is a positive number). 

Let’s try another example with a simple absolute value equation:

|2x – 3| = 7

To solve this equation, we again follow the two-case method. First, we need to consider both the positive and negative cases.

• Case 1: 2x – 3 = 7 
  • 2x = 10 
  • x = 5 
• Case 2: 2x – 3 = -7 
  • 2x = -4 
  • x = -2 

Thus, the solutions to the equation are:

x = 5 or x = -2

This process works for all absolute value equations that involve a single linear expression inside the absolute value bars.

Solving More Complex Absolute Value Equations

While the basic method for solving absolute value equations is useful for simple problems, more complicated absolute value equations may involve algebraic expressions with multiple terms inside the absolute value bars. In these cases, it’s important to follow the same two-case method but apply it carefully to the expression within the absolute value.

Let’s look at a more complex example:

|3x – 5| = 10

We solve this by considering both the positive and negative possibilities:

• Case 1: 3x – 5 = 10 
  • 3x = 15 
  • x = 5 
• Case 2: 3x – 5 = -10 
  • 3x = -5 
  • x = -5/3 

Thus, the solutions to this equation are:

x = 5 or x = -5/3

Why Two Cases?

The reason we need to consider both cases—positive and negative—when solving absolute value equations is that the absolute value represents the distance of a number from zero, and this distance is always positive. Therefore, the number inside the absolute value bars could either be positive or negative and still result in the same absolute value.

Special Considerations: Absolute Value Equations with Variables on Both Sides

In some cases, you may encounter an absolute value equation where the absolute value expressions are on both sides of the equation. This requires a slightly different approach. Let’s explore how to solve these types of equations.

For example, consider the equation:

|x – 4| = |2x + 1|

Here, both sides of the equation involve absolute values, so we need to apply the two-case method twice. We start by breaking down the absolute values on both sides:

• Case 1: x – 4 = 2x + 1 
  • x – 2x = 1 + 4 
  • -x = 5 
  • x = -5 
• Case 2: x – 4 = -(2x + 1) 
  • x – 4 = -2x – 1 
  • x + 2x = -1 + 4 
  • 3x = 3 
  • x = 1 
• Case 3: -(x – 4) = 2x + 1 
  • -x + 4 = 2x + 1 
  • -x – 2x = 1 – 4 
  • -3x = -3 
  • x = 1 
• Case 4: -(x – 4) = -(2x + 1) 
  • -x + 4 = -2x – 1 
  • x = -5 

The solutions are x = 1 and x = -5.

Important Tips for Solving Absolute Value Equations

While solving absolute value equations might seem straightforward, there are a few things you should keep in mind to avoid mistakes:

1. Always consider both the positive and negative cases when you have an absolute value equation. If the equation has more than one term inside the absolute value bars, break it down carefully for both positive and negative possibilities. 
2. Simplify the expression before solving if the equation involves a complex algebraic expression inside the absolute value. For example, in equations like |3x – 5| = 10, first solve the equation inside the absolute value, then apply the two-case process. 
3. Check for extraneous solutions. Sometimes, when solving absolute value equations, you may end up with solutions that don’t satisfy the original equation. Always plug your solutions back into the original equation to ensure they work. 
4. Be cautious of signs. When negating an expression inside the absolute value, make sure to flip the sign of the entire expression, not just the coefficient or constant term. Failing to do this can lead to incorrect solutions. 

Solving Absolute Value Inequalities

In addition to equations, absolute value inequalities are another common type of question you may encounter on the GRE. Solving absolute value inequalities follows a similar approach to solving equations, but there’s an important difference: when you multiply or divide by a negative number while solving inequalities, you must flip the inequality sign.

Let’s take a look at an example:

|2x – 3| < 5

We solve this inequality by considering both the positive and negative cases:

• Case 1: 2x – 3 < 5 
  • 2x < 8 
  • x < 4 
• Case 2: -(2x – 3) < 5 
  • -2x + 3 < 5 
  • -2x < 2 
  • x > -1 (Remember, we flip the inequality when dividing by a negative number.) 

Thus, the solution to the inequality is:

-1 < x < 4

This shows that x is between -1 and 4, exclusive.

Absolute Value and GRE Comparison Problems

Another common way that absolute value appears on the GRE is in comparison problems. In these problems, you are asked to compare two quantities involving absolute values and determine which one is larger. These problems can be tricky, as they may involve both positive and negative values. Let’s look at an example:

If |n| < 5 and |m| > 5, which is greater: |n| or |m|?

• Since |n| is less than 5, the absolute value of n must be some number between 0 and 5. 
• Since |m| is greater than 5, the absolute value of m must be some number greater than 5. 

Thus, |m| is greater than |n|. The answer is that Quantity B (|m|) is greater.

Mastering Absolute Value Equations

Solving absolute value equations and inequalities is a key skill for the GRE Quantitative section. While the concept of absolute value itself is relatively simple, it’s essential to understand how to apply it to equations, inequalities, and comparison problems. By following the systematic two-case approach for equations, remembering to flip the inequality sign when necessary, and checking for extraneous solutions, you can confidently tackle any absolute value question that comes your way on the GRE.

Advanced Applications of Absolute Value Equations: Dealing with Multiple Terms and Inequalities

In the previous sections, we covered the basics of solving absolute value equations and inequalities. Now, we will focus on more complex applications of absolute value, particularly when equations involve multiple terms or algebraic expressions inside the absolute value bars. We will also explore strategies for solving absolute value inequalities, which are commonly tested on the GRE. Understanding these advanced applications will help you feel confident and prepared for the most challenging absolute value problems on the test.

Solving Absolute Value Equations with Multiple Terms

When dealing with absolute value equations that contain more than one term inside the absolute value bars, the two-case method becomes especially important. However, these types of problems require careful attention to the algebraic steps involved. Let’s walk through a couple of examples to demonstrate the process.

Example 1: Equation with a Linear Expression Inside the Absolute Value

Consider the equation:

|3x – 7| = 12

To solve this equation, we apply the two-case method, which requires us to handle both the positive and negative cases.

• Case 1: 3x – 7 = 12 
  • 3x = 19 
  • x = 19/3 
• Case 2: 3x – 7 = -12 
  • 3x = -5 
  • x = -5/3 

Thus, the solutions to this equation are:

x = 19/3 or x = -5/3

Notice that in this example, the algebraic steps are relatively simple, but we must be careful to correctly solve for x in both cases.

Example 2: Equation with a More Complicated Expression Inside the Absolute Value

Let’s now tackle an equation that involves multiple terms inside the absolute value bars. For example:

|4x – 3| + 5 = 17

To solve this, we first isolate the absolute value term:

4x – 3 = 12

Now, we solve for x by following the two-case process:

• Case 1: 4x – 3 = 12 
  • 4x = 15 
  • x = 15/4 
• Case 2: 4x – 3 = -12 
  • 4x = -9 
  • x = -9/4 

Thus, the solutions to this equation are:

x = 15/4 or x = -9/4

Again, we followed the same general approach, but with more complicated algebra involved inside the absolute value expression. It’s crucial to break down the equation step by step and carefully solve for the variable in both the positive and negative cases.

Solving Absolute Value Inequalities: An Overview

Absolute value inequalities are slightly different from absolute value equations. While equations tell us that two quantities are equal, inequalities involve ranges of values that satisfy certain conditions. There are two types of absolute value inequalities that you may encounter: absolute value less than (|x| < a) and absolute value greater than (|x| > a).

Solving Absolute Value Inequalities of the Form |x| < a

An inequality like |x| < a represents the condition that the distance between x and 0 on the number line is less than a. This means that x must lie between -a and a. Mathematically, this inequality is equivalent to:

-x < x < a

Let’s work through an example:

|x – 3| < 4

This inequality means that the distance between x and 3 is less than 4. To solve this, we first break it down into two inequalities:

• x 3 < 4 (the positive case) 
• -(x – 3) < 4 (the negative case) 

Let’s solve both inequalities:

1. x – 3 < 4 
   • x < 7 
2. -(x – 3) < 4 
   • -x + 3 < 4 
   • -x < 1 
   • x > -1 

Thus, the solution is:

-1 < x < 7

This means that x must lie between -1 and 7.

Solving Absolute Value Inequalities of the Form |x| > a

On the other hand, an inequality like |x| > a represents the condition that the distance between x and 0 is greater than a. This means that x must lie outside the interval from -a to a. Mathematically, this inequality is equivalent to:

x < -a or x > a

Let’s solve an example:

|x + 2| > 5

This inequality tells us that the distance between x and -2 is greater than 5. To solve it, we break it into two cases:

• x + 2 > 5 (the positive case) 
• -(x + 2) > 5 (the negative case) 

Now let’s solve both inequalities:

1. x + 2 > 5 
   • x > 3 
2. -(x + 2) > 5 
   • -x – 2 > 5 
   • -x > 7 
   • x < -7 

Thus, the solution is:

x < -7 or x > 3

This means that x must be either less than -7 or greater than 3.

Summary of Absolute Value Inequalities

To summarize, solving absolute value inequalities follows these steps:

• For |x| < a: Split the inequality into two cases, where the expression inside the absolute value is less than a and greater than -a. 
• For |x| > a: Split the inequality into two cases, where the expression inside the absolute value is less than -a or greater than a. 
• Be sure to flip the inequality sign when multiplying or dividing by a negative number. 

Dealing with Absolute Value in GRE Comparison Problems

In addition to straightforward absolute value equations and inequalities, you may encounter comparison problems involving absolute value on the GRE. These questions require you to compare two quantities and determine their relationship. Absolute value comparison questions often ask you to determine which of two quantities is larger, or if they are equal. Let’s look at an example:

If |x| = 3 and |y| = 5, which quantity is larger?

In this case, the absolute value of x is 3, and the absolute value of y is 5. Since 5 is greater than 3, the answer is clear: Quantity B (|y|) is larger.

While comparison problems may seem simple, they often require a keen understanding of the properties of absolute value and careful attention to detail. It’s important to remember that absolute value represents distance, and you should focus on comparing the distances represented by the given values.

Advanced Techniques for Handling Absolute Value Questions

In more complex absolute value problems, you may need to apply additional strategies to find the solution. Here are some advanced tips to handle these types of questions:

Simplify the Equation First: In many problems, absolute value equations are complicated by additional terms or coefficients. Start by isolating the absolute value expression before applying the two-case method. This will make the problem easier to handle.

 

Consider the Context: Some absolute value problems may come with additional constraints or context that affect how you interpret the solution. Be sure to carefully read the problem and understand what is being asked.

 

Check for Extraneous Solutions: When solving absolute value equations, it’s always a good idea to plug your solutions back into the original equation to check for extraneous solutions. This can happen if you make algebraic errors during the process.

 

Be Careful with Inequalities: When solving absolute value inequalities, always be mindful of the inequality signs. Remember to flip the sign when multiplying or dividing by a negative number, as this can change the direction of the inequality.

 

Mastering Advanced Absolute Value Problems

By now, you should have a strong understanding of how to solve absolute value equations and inequalities, both simple and complex. The key to success is to break down the problems into manageable steps, using the two-case method for equations and carefully applying the rules for inequalities. Whether you’re working with linear expressions, quadratic equations, or inequalities, the strategies outlined in this section will help you navigate the more challenging GRE questions involving absolute value. Keep practicing, and you’ll find that these types of problems become much more intuitive and easier to solve.

Avoiding Common Mistakes and Mastering Absolute Value Problems

In the final part of our guide, we’ll tackle some of the most common mistakes that students make when solving absolute value problems on the GRE. Understanding these pitfalls will help you avoid errors and improve your performance. Additionally, we will review some advanced tips and provide a final set of practice problems to help reinforce the concepts.

Common Mistakes to Avoid

When working with absolute value equations, inequalities, or comparison problems, several common mistakes can trip up even experienced test-takers. Let’s go over these mistakes in detail so you can avoid them.

1. Forgetting to Consider Both Positive and Negative Solutions

One of the most frequent mistakes when solving absolute value equations is forgetting to consider both the positive and negative cases. Recall that absolute value represents distance, and the expression inside the absolute value bars can take both positive and negative values.

For example, consider the equation:

|x – 5| = 7

To solve, you must break it into two cases:

• Case 1: x – 5 = 7 → x = 12 
• Case 2: x – 5 = -7 → x = -2 

Both x = 12 and x = -2 are solutions, and failing to include both would result in an incomplete answer.

In GRE problems, this mistake can easily lead to wrong answers, especially when the question asks for all possible solutions. Always remember that if you have an absolute value equation like |x| = a, the solutions are both x = a and x = -a.

2. Misinterpreting Absolute Value Inequalities

Another common mistake occurs when solving absolute value inequalities, particularly those of the form |x| < a or |x| > a. It’s important to apply the correct logic when translating these inequalities into their equivalent forms.

• For |x| < a, the solution should be the interval (a, a), which means x must lie between -a and a. 
• For |x| > a, the solution should be two separate intervals: (-∞, -)-a and (a, ∞), meaning x must be less than -a or greater than a. 

Many students mistakenly reverse the logic when translating the inequality, leading to incorrect solutions.

For instance, solving the inequality:

|x + 3| < 5

You would break this down into two inequalities:

• x + 3 < 5 → x < 2 
• -(x + 3) < 5 → -x – 3 < 5 → -x < 8 → x > -8 

Thus, the correct solution is -8 < x < 2.

If you fail to properly handle the inequality, you might mistakenly end up with an incorrect range for x.

3. Not Isolating the Absolute Value Expression

When solving absolute value equations with additional terms, it’s critical to isolate the absolute value expression first before applying the two-case method. Failing to do so can make the equation much more complicated than necessary.

For example, consider this equation:

5 + |2x + 1| = 15

First, isolate the absolute value expression:

|2x + 1| = 10

Then, apply the two-case method:

• Case 1: 2x + 1 = 10 → x = 9/2 
• Case 2: 2x + 1 = -10 → x = -11/2 

By isolating the absolute value expression before solving, you simplify the problem and avoid unnecessary errors.

4. Neglecting to Check Solutions for Extraneous Results

It’s always a good practice to plug your solutions back into the original equation or inequality to check for extraneous solutions. This is particularly important when dealing with absolute value equations that involve quadratic or higher-degree terms.

For example, if you solve the equation:

|x^2 – 4| = 5

You might find solutions x = 3 or x = -3. However, after substituting back into the original equation, you might discover that one of the solutions does not satisfy the equation, particularly if there are algebraic manipulations that could introduce extraneous solutions.

Always verify your solutions before finalizing your answer, especially when solving more complex equations.

5. Misunderstanding the Relationship Between Absolute Value and Distance

One concept that often confuses is the relationship between absolute value and distance. The absolute value measures the distance between a number and zero on the number line. In some cases, students might misinterpret the absolute value expression in a way that overlooks this fundamental idea.

For instance, in the equation:

|x – 5| = 3

This means the distance between x and 5 is 3. Therefore, the solutions are:

x = 8 and x = 2

If you incorrectly interpret this as a simple linear equation, you might fail to find both solutions. Remember that absolute value represents distance, and consider both the positive and negative distances from the specified point.

Advanced Tips for Tackling Absolute Value Problems

Now that we’ve covered common mistakes, let’s move on to some advanced tips that can help you tackle more difficult absolute value problems on the GRE.

1. Break Down Complex Expressions

When the absolute value expression involves complex algebraic terms, such as polynomials or rational expressions, break the equation down step by step. Take the time to isolate the absolute value expression and simplify the terms before applying the two-case method.

For example, in an equation like:

|x^2 – 2x – 8| = 4

Start by factoring the quadratic expression:

(x – 4)(x + 2) = 4

Next, break it down into two cases:

• Case 1: x^2 – 2x – 8 = 4 
• Case 2: -(x^2 – 2x – 8) = 4 

By breaking down the equation into manageable steps, you will make it easier to solve without missing important algebraic manipulations.

2. Pay Attention to Coefficients and Constants

If the absolute value equation includes coefficients or constants, be extra cautious when solving. Make sure to distribute correctly and maintain the integrity of the equation. For example:

|2x – 5| + 3 = 12

First, isolate the absolute value expression:

|2x – 5| = 9

Then, solve using the two-case method:

• Case 1: 2x – 5 = 9 → x = 7 
• Case 2: 2x – 5 = -9 → x = -2 

By being attentive to the coefficients and constants, you avoid algebraic errors and ensure that you reach the correct solution.

3. Practice with a Variety of Problems

Finally, the best way to master absolute value problems is through practice. The GRE tests a variety of problem types, including simple equations, comparison problems, inequalities, and more complex expressions. The more practice problems you solve, the better you’ll understand how to approach different types of absolute value questions.

Additionally, when practicing, focus on time management. The GRE is a timed test, and some of the more difficult absolute value questions can be time-consuming. With sufficient practice, you’ll develop strategies to solve these problems more quickly and efficiently.

Practice Problems

Here are a few practice problems to test your understanding of absolute value equations, inequalities, and comparison questions:

1. Solve the equation: |2x + 3| = 7 
2. Solve the inequality: |x – 4| < 6 
3. Compare the quantities: 
   • Quantity A: |x – 5| 
   • Quantity B: 10
     If x = 3, which quantity is greater? 
4. Solve the equation: |x^2 – 4x| = 8 
5. Solve the inequality: |3x + 2| > 5 

Conclusion: Mastery of Absolute Value Problems on the GRE

By understanding the foundational principles of absolute value, practicing the two-case method for equations, and mastering the techniques for solving inequalities and comparison problems, you will be well-prepared for any absolute value question on the GRE. Keep in mind the common mistakes and advanced tips outlined in this guide, and apply these strategies consistently during your preparation.

Remember that success on the GRE comes down to mastering not just individual topics but also strategies that will help you efficiently solve problems within the time constraints of the exam. Absolute value may seem challenging at first, but with practice, you’ll gain the confidence to tackle these problems with ease and accuracy.